In "Quantum Field Theory" by Mark Srednicki, Chapter 9 page 67, after he proves that $\langle 0|\phi(x)|0 \rangle$ vanishes (meaning sum of all connected diagrams with a single source is zero), he makes the following claim:
Consider now that same infinite set of diagrams, but replace the single source in each of them with some other subdiagram. Here is the point: no matter what this replacement subdiagram is, the sum of all these diagrams is still zero. Therefore we need not bother to compute any of them! The rule is this: ignore any diagram that, when a single line is cut, falls into two parts, one of which has no sources. All of these diagrams (known as tadpoles) are canceled by the $Y$ counterterm, no matter what subdiagram they are attached to.
The most important question that I am wondering is how he arrived at this conclusion from his proof that the $Y$ counterterm can be used to make $\langle0|\phi(x)|0\rangle$ zero.
Also, what does he mean by "subdiagram" here? A part of the diagram formed by cutting one of the diagram that has a source, or a part of any diagram that does not necessarily have a source? Is he replacing each of the different diagrams with single source with identical subdiagram or replacing each source with different subdiagrams? (Since "subdiagram" is singular, I am guessing they are all replaced with identical subdiagrams.)